COMPLEX GEOMETRY Course notes

2.5 Dimension on Riemann surfaces
Definition 2.5.1. A divisor D on a Riemann surface S is a formal Z-linear combination of points in S
D = ∑ a i P i
where a i ≠ 0 for every i and {P i } is a discrete subset of S. A divisor D is called effective if a i ≥ 0 for every i.
If supp(D) = {P i / a i ≠ 0} is finite, then
deg(D) := ∑ a i
Definition 2.5.2. Let f ∈ M(S) − {0} = M ∗ (S). The divisor of f is defined by
(f) := (f) 0 − (f) ∞
where
(f) 0 = ∑ (ord P f)P and (f) ∞ = ∑ P ∈f −1 (∞) (mult P f)P
Note that mult P f = −ord P f, so we can rewrite the previos expresion as
(f) = ∑ (ord P f)P
Lemma 2.5.1. If S is a compact Riemann surface, then deg(f) = 0 for every f ∈ M ∗ (S), where deg is a
map Div(S) −→ Z.
Definition 2.5.3. A divisor is called principal if it lies in the image of deg( ) : M ∗ −→ Z.
Definition 2.5.4. Two divisors D 1 and D 2 are said to be linearly equivalent, denoted D 1 ∼ D 2 , if D 1 −D 2
is principal.
Example 2.5.1. D 1 ∼ D 2 on P 1 if and only if deg(D 1 ) = deg(D 2 ).
Example 2.5.2. What condition we need if we want D 1 ∼ D 2 on C = C/Γ. Let p, q ∈ C be two distinct
points in C and suppose that D = p − q = (f) for some f ∈ M ∗ . Then f is a map C −→ P 1 with deg(f) = 1.
We have (f) 0 = p and f is bijective. Then f is a biholomorphic map, getting a contradiction.
Given ω ∈ M ′ (S) ∗ . Recall that this means ω = fdz for a local coordinate z at p and f ∈ M(p).
Definition 2.5.5. ord p ω = ord p f. The divisor of the form
(ω) := ∑ p∈S (ord pω)P
is called a canonical divisor and is denoted K S or simply K. A divisor is canonical if D ∼ (w).
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